Abstract: It is shown how the elliptic finite-gap potentials of the Schrödinger equation give rise to a family of solvable linear differential equations of the Fuchs class on the plane and on the torus: the latter case cannot be integrated via realizations of the Zinger-Kovacic type algorithms known in the Picard-Vessiot theory. For the arising Fuchsian equations, monodromy groups and their representations are constructed, the differential Galois group is described, together with a (recursive) method for calculation of the objects involved therein.

12.E.
T. Whittaker and G.
N. Watson, A course of modern analysis, Cambridge Mathematical
Library, Cambridge University Press, Cambridge, 1996. An introduction to
the general theory of infinite processes and of analytic functions; with an
account of the principal transcendental functions; Reprint of the fourth
(1927) edition. MR
1424469